Manuscript VII-F-ST@1 · VII-F-ST.tex · theorem · 9eee7a44

Theorem: Winding-marked affine word

Adapted affine charts and framed peripheral holonomy · lines 795–813 · proof included or linked

VII-F-ST@1#thm:word

For a closed word $f_{0}, \dots, f_{m} = f_{0}$ ,

g_{f_{0} \leftarrow f_{m - 1}} \dots g_{f_{1} \leftarrow f_{0}} = A_{f_{0}} J^{N} A_{f_{0}}^{- 1}, N = j = 0 \sum^{m - 1} n_{f_{j + 1} f_{j}} .

For a realized loop avoiding $W$ , $N$ is its oriented transverse winding number. Every small regular vertex loop has affine product $I$ .

Proof. Substitution cancels adjacent chart inverses, leaving the powers of $J$ . Lifting the spatial loop to the angular cover changes its lifted angle by $N (3 π /2)$ , which identifies the winding. Small regular loops have zero winding.

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unlabeled · line 798
$g_{f_{0} \leftarrow f_{m - 1}} \dots g_{f_{1} \leftarrow f_{0}} = A_{f_{0}} J^{N} A_{f_{0}}^{- 1}, N = j = 0 \sum^{m - 1} n_{f_{j + 1} f_{j}} .$

Theorem: Winding-marked affine word

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